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Chapter 4 Trigonometry Course/Section Lesson Number Date Section 4.4 Trigonometric Functions of Any Angle Section Objectives: Students will know how to evaluate trigonometric functions of any angle and of real numbers. I. Introduction (pp. 312−313) Pace: 10 minutes State the following definitions of trigonometric functions of any angle. Let be an angle in standard position with (x, y) a point on the terminal side of and r x 2 y 2 0. sin tan sec y r y ,x x r ,x x cos 0 cot 0 csc x r x ,y 0 y r ,y 0 y Example 1. Let (4, −3) be on the terminal side of . Find the value of the sine, cosine, and tangent of . r 4 2 3 2 5 . So, sin = −3/5, cos = 4/5, and tan = −3/4. Discuss the signs of the trigonometric functions in the four quadrants. Because both x and y are positive in the first quadrant, all six functions are positive in the first quadrant. Because only y is positive in the second quadrant, only sine and cosecant are positive in the second quadrant. Since both x and y are negative in the third quadrant, only tangent and cotangent are positive in the third quadrant. Since only x is positive in the fourth quadrant, only cosine and secant are positive in the fourth quadrant. Example 2. Given cos = 3/5 and tan < 0, find sin and cot . must be in the fourth quadrant. Using x = 3, y = −4, and r = 5, we have sin = −4/5 and cot = −3/4. II. Reference Angles (p. 314) Pace: 5 minutes State that if is in standard position, then the reference angle associated with is the acute angle formed by the terminal side of and the x-axis. Tip: Emphasize that the reference angle uses only the x-axis. Example 3. Find the reference angle for the following angles. a) = 125 = 180 − 125 = 55 b) =5 =2 −5 1.2832 III. Trigonometric Functions of Real Numbers (pp. 315−317) Pace:15 minutes State that if is in standard position with (x, y) on its terminal side, then if is placed in standard position, (|x|, |y|) will be on its terminal side. Also, the r for and the r for will be the same. Hence, |sin | = sin , |cos | = cos , and |tan | = tan . This means that to evaluate trigonometric functions of any angle, we need only find the value of that function for the Larson/Hostetler Precalculus with Limits Instructor Success Organizer Copyright © Houghton Mifflin Company. All rights reserved. 4.4-1 reference angle and attach the proper sign, according to the quadrant in which lies. 4.4-2 Larson/Hostetler Precalculus with Limits Instructor Success Organizer Copyright © Houghton Mifflin Company. All rights reserved. Tell the students that they should memorize the table on page 315 of the text. Example 4. Evaluate the following trigonometric functions. a) sin 210 . Since 210 is in quadrant III and the reference angle is 30 , sin 210 = −sin 30 = 1/2. b) cos (−5 /4). Since −5 /4 is in the second quadrant and the 5 2 reference angle is /4, cos . cos 4 2 4 c) tan 5 /3. Since 5 /3 is in the fourth quadrant and the 5 reference angle is /3, tan tan 3. 3 3 Example 5. Let be an angle in the third quadrant such that cos −1/4. Find the following. a) sin 1 2 sin 2 1 4 15 sin 2 16 sin = 15 4 b) tan tan sin cos 15 4 14 15 Example 6. Use a calculator to solve sin = 0.2358. SIN-1 .2358 ENTER . 0.2380 Larson/Hostetler Precalculus with Limits Instructor Success Organizer Copyright © Houghton Mifflin Company. All rights reserved. 4.4-3